No you probably don;t want to use ASIO with a realtec. Probably wouldn't work anyway.
Of course, the way you are recording isn't idea. Especially using a factory card with synths and midi..... MME generally has bad latency with that setup. The internal mic could very well be the source of the issue.
Since the sound comes through on the bounce, that means it not a spurious noise. It is actually being recorded to the track. Any number of things could be causing it. Feedback, some sort of issue in the sound card/driver mode, or something else entirely.
Since you are new here, let me ask you a few questions.
Are you considering getting into home recording as a relatively serious hobby or possibly as a career musician?
Some folks are just dabbling and others are musicians who play and want to record at home. The reason I ask, is, if you are anywhere close to being serious about home recording for fun or profit, I would suggest an investment in a few things.
1. A decent audio/midi interface which runs native ASIO and has phantom power and connects with USB.
2. A decent condenser mic
3. A pair of inexpensive but decent powered studio monitors
4. A basic pair of headphones.
That is the beginnings to a decent home studio. A nice interface eliminates the feedback issue, also eliminates the MME/latency and midi issues, and external speakers moves the speakers out of the computer as does the mic.
Before I close could you post the track with the offending noise to soundclick or soundcloud so we can hear what you are hearing? Doing that may help us to pinpoint the issue. Also any screen shots of your setup preferences for the inputs and outputs to those tracks would be useful as well.